Re: Groups defined by as simple as possible rules... as computer strings?

On Sun, 19 Jun 2005 13:57:23 +0200, Anton Suchaneck
<asuchaneck@xxxxxx> wrote:

>Hi,
>
>I've just learned about group theory and I was wondering if it is possible
>to define specific groups by a few equations only.
>
>For example a simple cyclic group could be defined by g*g*g*g=1 and all
>other elements just can be written as g*g and g*g*g without giving them
>explicit names.
>
>Of course sometimes these defining equations contain more than one element
>and it would be good to minimize the number of elements used.
>
>Is there a method to do this systematically and is it possible to use
>equations of the form "...=1" only? (say a*b*c=1 and b*b=1)
>
>I'd find it instructive to imagine groups as computer strings (e.g. "gg",
>"ggg") which can be catenated with rules of cancellation (e.g.
>"gg"+"ggg"="ggggg"="g" since "gggg"=""). In this model the associative
>property is implicit.
>
>A problem that *shouldn't* appear in the rules is that one had to expand a
>string to reduce it again, e.g.
>reduce "abc" with "bc=aac", "aaa="":
>so that "abcd"="aaacc"="cc"
>
>Is that possible?

What you're talking about here is more or less a standard
thing, describing a group in terms of "generators and
relations". That's a perfectly good way to describe a lot
of groups.

Unfortunately it doesn't work as well as you might hope.
The expansion you say you want to avoid cannot always be
avoided. And in fact there's a thing called the "unsolvability
of the word problem": if you have a group described by
generators and relations as above there is no algorithm
to determine whether two strings ("words") are equal.

>Anton


************************

David C. Ullrich
.

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